tag:blogger.com,1999:blog-264226589944705290.post5431868449333115997..comments2018-07-08T12:16:00.281-07:00Comments on God Plays Dice: geographical random walksMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-264226589944705290.post-89499030672763106482007-08-04T20:59:00.000-07:002007-08-04T20:59:00.000-07:00Should two way streets be counted as two streets c...Should two way streets be counted as two streets coming into the intersection, and one way streets be counted as one? I think so, but perhaps a two way street is not worth quite as much as two one-ways at an angle. In fact, I wonder if you should multiply the streets by something like c + (1-c)sin(alpha), where c is some constant in [0,1], maybe 1/2, and alpha is the angle between each street in the intersection and the next one clockwise. (In the case of only two streets, use the smallest angle, so alpha is always in [0, pi/2].) I suggest this because an intersection between streets that are at a greater angle may be more likely to bring together travellers from places that are further apart, and may thus serve a wider area.<BR/><BR/>But ignoring these considerations, the general approach you mention -- analyzing the intersections -- is essentially the same as the Google algorithm for page rank, where links like intersections and pages are like streets. Indeed, one way of thinking about page rank is to think of it as a rough reading of the likelihood you'd wind up on a particular page by browsing somewhat randomly. And, analogously to your considerations of weighing some streets more heavily than others, Google weighs some links more heavily than others, based on position on the page and based on fonts used, as well as some other factors.<BR/><BR/>The computation involves reading out the eigenvector of the square adjacency matrix, where the entry for each page gives page rank, and pages are adjacent if there is a link between them. (But there are some manipulations needed to ensure that this can be found and is stable. The matrix is transformed to make it an irreducible, stochastic matrix.) The AMS published <A HREF="http://www.ams.org/featurecolumn/archive/pagerank.html" REL="nofollow"><BR/>a nice article on it.<BR/></A><BR/>Google also takes some measures to identify completely "fake" links, which exist only to boost page rank. Of course there's no equivalent problem with roads.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-25226398953275478912007-08-04T15:23:00.000-07:002007-08-04T15:23:00.000-07:00You might guess from a map that Baltimore and Broa...You might guess from a map that Baltimore and Broadway are important streets, because in cities built in grids, diagonal streets are more likely to be used to minimize walking distance (or alternately, are so old that they predate the grid pattern).Melaniehttps://www.blogger.com/profile/05385756739431742663noreply@blogger.com